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A003701 Expansion of e.g.f. exp(x)/cos(x).

%I A003701 M1259 #75 May 12 2024 17:42:30
%S A003701 1,1,2,4,12,36,152,624,3472,18256,126752,814144,6781632,51475776,
%T A003701 500231552,4381112064,48656756992,482962852096,6034272215552,
%U A003701 66942218896384,929327412759552,11394877025289216,174008703107274752,2336793875186479104,38928735228629389312
%N A003701 Expansion of e.g.f. exp(x)/cos(x).
%C A003701 Binomial transform of A000364 (with interpolated zeros). Hankel transform is A055209. - _Paul Barry_, Jan 12 2009
%D A003701 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003701 Alois P. Heinz, <a href="/A003701/b003701.txt">Table of n, a(n) for n = 0..485</a> (first 101 terms from T. D. Noe)
%H A003701 T. Chow, Fair permutations and random k-sets, <a href="https://www.jstor.org/stable/10.4169/000298910x515820">Problem 11523</a>, Amer. Math. Monthly 117 (October 2010), 741; <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.119.09.800">solution</a> by Jim Simons, Amer. Math. Monthly 119 (November 2012), 801-803.
%H A003701 J. W. Layman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%F A003701 G.f.: 1/(1-x-x^2/(1-x-4x^2/(1-x-9x^2/(1-x-16x^2.... (continued fraction). - _Paul Barry_, Jan 12 2009
%F A003701 E.g.f.: exp(x)*sec(x). - _Zerinvary Lajos_, Apr 05 2009
%F A003701 E.g.f.: 1+x/H(0); H(k)=4k+1-x+x^2*(4k+1)/((2k+1)*(4k+3)-x^2+x*(2k+1)*(4k+3)/(2k+2-x+x*(2k+2)/H(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Nov 15 2011
%F A003701 G.f.: 1/G(0) where G(k)= 1 - 2*x*(k+1)/(1 + 1/(1 + 2*x*(k+1)/G(k+1))); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Nov 20 2012
%F A003701 G.f.: -1/x/Q(0), where Q(k)= 1 - 1/x - (k+1)^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Apr 26 2013
%F A003701 G.f.: (1-x)/Q(0), where Q(k)= (1-x)^2 - (1-x)^2*x^2*(k+1)^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 04 2013
%F A003701 a(n) ~ n! * ((-1)^n*exp(-Pi/2) + exp(Pi/2)) *(2/Pi)^(n+1). - _Vaclav Kotesovec_, Oct 08 2013
%F A003701 G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)/( x*(2*k+1) - 1/(1 + x*(2*k+1)/( x*(2*k+1) + 1/(1 - x*(2*k+2)/( x*(2*k+2) - 1/(1 + x*(2*k+2)/( x*(2*k+2) + 1/Q(k+1) ))))))); (continued fraction). - _Sergei N. Gladkovskii_, Oct 22 2013
%F A003701 G.f.: Q(0)/(1-x), where Q(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 - (1-x)^2/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 21 2013
%e A003701 G.f. = 1 + x + 2*x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 152*x^6 + 624*x^7 + 3472*x^8 + ...
%p A003701 G(x):= exp(x)*sec(x): f[0]:=G(x): for n from 1 to 54 do f[n]:= diff(f[n-1],x) od: x:=0: seq(f[n], n=0..22); # _Zerinvary Lajos_, Apr 05 2009
%p A003701 # second Maple program:
%p A003701 b:= proc(u, o) option remember;
%p A003701       `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
%p A003701     end:
%p A003701 a:= n-> add(`if`(j::odd, 0, b(j, 0)*binomial(n, j)), j=0..n):
%p A003701 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 12 2024
%t A003701 a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ x ] / Cos[x], {x, 0, n}]] (* _Michael Somos_, Jun 06 2012 *)
%o A003701 (PARI) x='x+O('x^66); Vec(serlaplace(exp(x)/cos(x))) \\ _Joerg Arndt_, May 07 2013
%o A003701 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)/Cos(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Oct 14 2018
%Y A003701 Cf. A009739, A062272, A062161.
%Y A003701 Bisections are A000795 and A002084.
%K A003701 nonn
%O A003701 0,3
%A A003701 _R. H. Hardin_
%E A003701 Extended and reformatted 03/97.